Layer 3 · the derivation · Dark matter identity

γ_PPN = β_PPN = 1, by two independent routes

Branch-B static-fluid theorem (F128) and the Palatini-class PPN result under path ζ (F130b). Both yield the GR value at leading order; corrections scale as O(f_s,☉).

Result

γ_PPN = β_PPN = 1 at leading order, by two independent routes.

Branch-B static-fluid theorem (F128) and Palatini-class PPN under path ζ (F130b). Higher-order corrections scale as O(f_s,☉); Cassini compatibility is conditional on the solar interior's structural index.

Cassini bound: |γ − 1| ≤ 2.3 × 10⁻⁵ · ISST: γ = 1 + O(f_s,☉)

Step 1

Start from the F01 unified action

S_{t e x t I S S T} = in t B i g [, t f r a c P s i 16 p i, R; +; (1 + f), ma t h c a l L_{m}, B i g] s q r t - g, d^{4} x

The single Lagrangian. Vary with respect to g^μν and Ψ to get the field equations. Two branches emerge from the Ψ-equation: Branch A (matter-trace sources Ψ) and Branch B (R = 0 with constant Ψ on regular sources).

Frameworks consulted

ALLOWED·F01 unified action

Step 2

Solar-system context: weak-field, slow-motion, asymptotically flat

P hi / c^{2} s im 1 0^{- 6}, q u a d v / c l l 1, q u a d g_{m u n u} t o e t a_{m u n u};; t e x t a t in f ini t y

Standard PPN preconditions hold for the Sun. Φ_☉/c² ≈ 2 × 10⁻⁶; planetary motion is non-relativistic; the metric is asymptotically Minkowski at scales ≪ H₀⁻¹. The PPN expansion is therefore the right tool here (it is not applicable to neutron stars or black holes, which are flagged separately on the master passport).

Frameworks consulted

ALLOWED·PPN framework
ALLOWED·Cassini γ bound
ALLOWED·LLR / Nordtvedt η

Step 3a

Route A — Branch-B static-fluid theorem

\\nabla_\\mu \\Psi = 0 \\;\\;\\Longleftrightarrow\\;\\; R = 0\\;\\;\\text{(static, perfect fluid, regular at origin)}

F128 establishes that the only regular solution for a static, spherically symmetric perfect-fluid source on Branch B has Ψ constant. With Ψ = Ψ₀ everywhere outside the source, the metric is exactly Schwarzschild to all orders, and $G_{t e x t e f f} = 1/ P s i_{0} = t e x t co n s t$ identically. The standard Schwarzschild PPN expansion gives γ = β = 1 with no corrections.

Frameworks consulted

ALLOWED·Branch-B static fluid
ALLOWED·Schwarzschild exterior
DENIED·Brans-Dicke[scalar_propagates=false]

Step 3b

Route B — Palatini-class PPN under path ζ

(1 + f_{p}) (1 + f_{s}), T = f r a c 1 8 p i, R, P s i

nab l a^{2} P hi = 4 p i, G_{t e x t e f f} (r h o), r h o, q q u a d G_{t e x t e f f} = G_{N} c d o t B i g [1 + O (f_{s}) B i g]

F130b §A.4 derives the PPN expansion under the corrected path-ζ constraint. The Ψ-equation is algebraic (Palatini-like): given the matter trace at a point, Ψ is determined locally with no propagation. The exterior is Schwarzschild with G_eff evaluated at the (vanishing) exterior matter density, giving γ = 1 at leading order. The next-order correction scales as the structural index f_s,☉ of the solar interior — expected ≲ 10⁻⁴ for thermalised matter at solar temperatures.

Frameworks consulted

ALLOWED·Palatini PPN (Olmo 2005)
DENIED·DEF PPN[scalar_propagates=false]
DENIED·Chameleon screening[constraint_class=palatini]

Step 4

Compare to Cassini

∣ g amma - 1∣ l e 2.3 t im es 1 0^{- 5};; t e x t (B er t o tt i, I ess, T or t or a 2003)

Both routes reproduce the standard Schwarzschild result at leading order. The Cassini bound is satisfied automatically by Route A (no correction at all to all orders considered); Route B requires f_s,☉ ≲ 10⁻⁴, which is a measurable quantity but expected to be small for hot, well-mixed solar plasma. This is a kill condition, not a fit parameter — if a future measurement of the solar structural index gives a value inconsistent with the Cassini bound, the path-ζ formulation fails.

Frameworks consulted

ALLOWED·Cassini Shapiro delay

Step 5

What this denies

Several alternative modified-gravity frameworks are denied at this step because the master passport stamps scalar_propagates = false and constraint_class = palatini. ISST passes Cassini not by tuning a screening mechanism but by structural property: the scalar carries no propagating degree of freedom.

Frameworks consulted

DENIED·Brans-Dicke PPN[scalar_propagates=false (no Yukawa correction)]
DENIED·DEF (Damour-Esposito-Farèse)[scalar_propagates=false]
DENIED·Chameleon screening[constraint_class=palatini]
DENIED·f(R) metric[dof_count=2, not 3]
DENIED·Vainshtein[constraint_class=palatini]